{"paper":{"title":"Double-jump phase transition for the reverse Littlewood--Offord problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Julien Portier, Lawrence Hollom, Victor Souza","submitted_at":"2025-03-31T15:20:49Z","abstract_excerpt":"Erd\\H{o}s conjectured in 1945 that for any unit vectors $v_1, \\dotsc, v_n$ in $\\mathbb{R}^2$ and signs $\\varepsilon_1, \\dotsc, \\varepsilon_n$ taken independently and uniformly in $\\{-1,1\\}$, the random Rademacher sum $\\sigma = \\varepsilon_1 v_1 + \\dotsb + \\varepsilon_n v_n$ satisfies $\\|\\sigma\\|_2 \\leq 1$ with probability $\\Omega(1/n)$. While this conjecture is false for even $n$, Beck has proved that $\\|\\sigma\\|_2 \\leq \\sqrt{2}$ always holds with probability $\\Omega(1/n)$. Recently, He, Ju\\v{s}kevi\\v{c}ius, Narayanan, and Spiro conjectured that the Erd\\H{o}s' conjecture holds when $n$ is odd."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.24202","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.24202/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}